# How do I introduce control variables in a regression where the I am comparing coefficients to the base?

I am running regressions where I need to compare the coefficients with the base, but I am a little confused about how to introduce the control variables.

My regression looks as per:

log(wage) = B + B(black) + B(Hispanic) + B(black*noeducation) +
B(Black*Highschool) + B(Black*College) + B(Hispanic*noeducation) +
B(Hispanic*Highschool) +B(Hispanic*College) + B(2002) + B(2003) + B(2004) +
B(education) + B(office) + B(Highschool) + B(College)


This is whereby my data set has the following variables:

Race variables : Black, Hispanic, White
Education variables: no education, High School, College
Year variables: 2001, 2002, 2003, 2004
Occupation variables: education, medical, office

The coefficients of interest are B(Hispanic$$*$$noeducation) and B(Black$$*$$noeducation) because I want to see if there is a difference for returns on wages for black people and hispanic people with no education compared to white people with no education. However, I am scared about my controls affecting it because technically won't these coefficients tell me the difference of return on wages of black people and hispanic people compared to white people in 2001, in the medical sector with no education. If this is correct, how would I overcome this to just compare regardless of year and occupation variables?

Your model can't be estimated the way it is. You have only 3 levels of eduction so you only need two main effects and two interaction terms with each level of race.

If you remove the education, Hispanic*noeducation and Black*noeducation terms then the coefficients for Black and Hispanic will estimate the quantities you are interested in. This is because when you include interaction terms, the main effect of race will estimate its effect at the reference levels of the other covariates for which there are interactions.

To see this, consider what the model predicts for a hispanic person with no education compared to a white person with no education (for any given year and sector, but suppose for 2002 in an office job):

$$\log{(\text{wage})}_{\text{Hispanic no education}} = B + B_{Hispanic} + B_{2002} + B_{office}$$

$$\log{(\text{wage})}_{\text{White no education}} = B + B_{2002} + B_{office}$$

So the difference is: $$B_{Hispanic}$$

Note this would work whatever year and sector. Because there is no interaction between race and year or race and sector in your models, those terms (for sector and year) will always cancel out. So long as the white person and the hispanic person were working in the same sector in the same year, the difference between their expected wages (under your model) would always be the same. Only if their education level varied would you see a different effect.